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A204113 Array:  row n shows the coefficients of the characteristic polynomial of the n-th principal submatrix of the matrix at A204112, given by f(i,j)=GCD(F(i+1),F(j+1)), where F=A000045 (Fibonacci numbers). 3
1, -1, 1, -3, 1, 2, -8, 6, -1, 8, -36, 35, -11, 1, 48, -232, 274, -116, 19, -1, 576, -2880, 3620, -1728, 358, -32, 1, 10368, -52992, 70632, -37192, 8906, -1016, 53, -1, 331776, -1716480, 2354112, -1294352, 332812, -42924, 2805 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Let p(n)=p(n,x) be the characteristic polynomial of the n-th principal submatrix.  The zeros of p(n) are real, and they interlace the zeros of p(n+1).  See A202605 and A204016 for guides to related sequences.

REFERENCES

(For references regarding interlacing roots, see A202605.)

LINKS

Table of n, a(n) for n=1..42.

EXAMPLE

Top of the array:

1....-1

1....-3....1

2....-8....6....-1

8....-36...35...-11...1

MATHEMATICA

u[n_] := Fibonacci[n + 1]

f[i_, j_] := GCD[u[i], u[j]];

m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]

TableForm[m[8]] (* 8x8 principal submatrix *)

Flatten[Table[f[i, n + 1 - i],

  {n, 1, 15}, {i, 1, n}]]    (* A204112 *)

p[n_] := CharacteristicPolynomial[m[n], x];

c[n_] := CoefficientList[p[n], x]

TableForm[Flatten[Table[p[n], {n, 1, 10}]]]

Table[c[n], {n, 1, 12}]

Flatten[%]                   (* A204113 *)

TableForm[Table[c[n], {n, 1, 10}]]

CROSSREFS

Cf. A204112, A202605, A204016.

Sequence in context: A060750 A204025 A204126 * A204128 A266272 A201677

Adjacent sequences:  A204110 A204111 A204112 * A204114 A204115 A204116

KEYWORD

tabl,sign

AUTHOR

Clark Kimberling, Jan 11 2012

STATUS

approved

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Last modified June 21 16:12 EDT 2018. Contains 305624 sequences. (Running on oeis4.)